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Approximations and errors — KCSE Mathematics

KCSE Mathematics · 106 practice questions · 3 syllabus objectives · 3 revision lessons

36 easy35 medium35 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Define absolute, relative, percentage, round-off and truncation errors

Determine possible errors from computations and find maximum and minimum errors

Express values to a given number of significant figures and make reasonable approximations

Revision Notes

Concise lesson notes for Approximations and errors, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Understanding Errors in Mathematics

In mathematics, it is crucial to understand different types of errors that can occur in calculations. Here are the key definitions:

  • Absolute Error: The difference between the true value and the measured value. It is expressed as:

    Absolute Error = |True Value - Measured Value|

  • Relative Error: This is the absolute error divided by the true value, often expressed as a fraction or percentage:

    Relative Error = Absolute Error / True Value

  • Percentage Error: This is the relative error expressed as a percentage:

    Percentage Error = (Relative Error) × 100%

  • Round-off Error: This occurs when a number is approximated to fewer decimal places, leading to a loss of precision.

  • Truncation Error: This happens when a mathematical procedure is stopped early (e.g., not using enough terms in a series expansion). It is the difference between the true value and the truncated value.

Understanding these concepts is essential for accurate data interpretation and reporting in mathematics.

Key points to remember

  • Absolute error measures the exact difference from the true value.
  • Relative error compares absolute error to the true value.
  • Percentage error expresses relative error as a percent.
  • Round-off error results from approximating numbers.
  • Truncation error occurs when a procedure is stopped early.

Worked example

Define absolute error and calculate it for a true value of 50 and a measured value of 48. Absolute Error = |50 - 48| = 2.

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More lessons in this topic

Lesson 2: Understanding Errors in Computations

Objective: Determine possible errors from computations and find maximum and minimum errors

In mathematics, approximations often lead to errors in calculations. It's essential to determine possible errors to ensure accuracy. Maximum error is the largest possible deviation from the true value, while minimum error is the smallest. To find these errors, follow these steps:

  1. Identify the approximate value (A) and the exact value (E).

  2. Calculate the absolute error using the formula:

    Absolute Error = |A - E|

  3. Determine the maximum error by considering the range of possible values around A.

  4. Find the minimum error by assessing the closest approximation to E.

For example, if a length is measured as 5.2 m (approximate value) and the true length is 5.0 m (exact value):

  • Absolute Error = |5.2 - 5.0| = 0.2 m
  • Maximum Error = 0.2 m (if no other errors are considered)
  • Minimum Error = 0.0 m (if A is exactly E)

Understanding these concepts helps in assessing the reliability of your computations.

  • Maximum error is the largest deviation from the true value.
  • Minimum error is the smallest deviation from the true value.
  • Calculate absolute error using |A - E|.
  • Consider the range of possible values for max error.
  • Minimum error occurs when the approximation is exact.

A student measures a length as 7.5 cm, but the true length is 7.3 cm.

  • Absolute Error = |7.5 - 7.3| = 0.2 cm.
  • Maximum Error = 0.2 cm, Minimum Error = 0.0 cm.
Lesson 3: Significant Figures and Approximations

Objective: Express values to a given number of significant figures and make reasonable approximations

In mathematics, expressing values to a given number of significant figures is essential for precision. Significant figures are the digits in a number that contribute to its accuracy. To express a number in significant figures:

  1. Identify non-zero digits: All non-zero digits are significant.
  2. Leading zeros are not significant.
  3. Captive zeros (zeros between non-zero digits) are significant.
  4. Trailing zeros in a decimal number are significant.

For example, to express 0.004560 in three significant figures, we identify:

  • The significant digits are 4, 5, and 6.
  • Thus, 0.004560 rounded to three significant figures is 0.00456.

When making reasonable approximations, consider the context of the problem. For instance, if you are estimating a total cost of items priced at Ksh 199.99 and Ksh 150.00, you might round them to Ksh 200 and Ksh 150 to simplify calculations. The total becomes Ksh 350.

In summary, mastering significant figures and approximations helps in achieving clarity and accuracy in mathematical computations.

  • Significant figures indicate accuracy in numerical values.
  • Leading zeros do not count as significant figures.
  • Captive zeros between non-zero digits are significant.
  • Round numbers to the required significant figures for clarity.
  • Use reasonable approximations for easier calculations.

Express 0.007890 to two significant figures.

  • Identify significant digits: 7 and 8.
  • Rounded value is 0.0079.

Sample Questions

Read 3 questions and answers free. Sign up to access all 106 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

A thermometer reads 37.0 °C to one decimal place. (a) State the maximum and minimum possible temperatures. (2 marks) (b) If the actual temperature is 36.8 °C, determine the percentage error. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Minimum possible temperature = 36.95 °C (1 mk)
Maximum possible temperature = 37.05 °C (1 mk)
Part (b) — 2 marks
Absolute error = |37.0 - 36.8| = 0.2 °C (1 mk)
Percentage error = (0.2 / 37.0) × 100 = 0.5405% (to 4 decimal places) (1 mk)
2
easySHORT ANSWER3 marks

A student measures the length of a table as 1.25 m accurate to two decimal places. (a) State the range of possible lengths for the table. (2 marks) (b) If the actual length is 1.23 m, calculate the absolute error. (1 mark)

Answer & marking scheme

Part (a) — 2 marks
Minimum possible length = 1.245 m (1 mk)
Maximum possible length = 1.255 m (1 mk)
Part (b) — 1 mark
Absolute error = |1.25 - 1.23| = 0.02 m (1 mk)
3
easySHORT ANSWER2 marks

A round table's diameter is recorded as 1.2 m, accurate to the nearest 0.1 m. Identify the absolute error in the measurement. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Absolute error = 0.05 m (1 mk)
This is half of the precision of the measurement (1 mk)
4

A piece of wood is measured to be 2.5 m long, accurate to the nearest 0.1 m. Identify the maximum and minimum possible lengths of the wood. (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Approximations and errors" cover?

Absolute, relative, percentage, round-off and truncation errors; propagation of errors

How many practice questions are available for Approximations and errors?

HighMarks has 106 Approximations and errors practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Approximations and errors for the KCSE exam?

Start with the revision notes on this page to refresh the core concepts, then work through the practice questions in increasing difficulty. Sign up for HighMarks to get a personalised study plan that adapts to the topics you keep getting wrong, plus mock exams, subject-wide practice, and detailed performance tracking. See pricing.

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